Lecture 23: MOI & Torque

1. Lecture 23: MOI & Torque

2. Kinetic energy of Rotation • K = sum of ½ m v2 for all parts of the body •  Moment of inertia I •  K = ½ I ω2

3. Example: Two objects connected with a massless rod  I • Mass of 1 kg at -2m, of 4kg at 1m •  I = 1kg (-2m)^2 + 4kg (1m)^2 = 8 kg m^2 • Different I around different axis! • Example: rotate around midpoint: I’ = 11.25 kg m^2

4. Post-lecture Exercise (10.1 – 10.6)   • Two masses of 2 kg are connected by a massless 1m rod and rotated around their center of mass with a period of 2s. Calculate the rotational kinetic energy of this configuration. • K = 9.87 J • Use Eqs. (10-33) & (10-34): I = 2kg(-0.5m)^2+ 2kg(0.5m)^2 = 1 kg m^2 ω = 2π/T = π Hz, so K = ½ I ω ^2 = ½ π^2 J

5. Determining MOI • Integral • Table • Parallel-axis theorem

6. Torque • Torque is force times lever arm • Lever arm is distance to rotation axis along a direction perpendicular to the force • Later:τ = r x F • | τ | = |r| |F| sin φ

7. Pre-lecture Exercise (10.7 – 10.10) • In the simulation Torque how large does the red force have to be (if the red position is negative 1m and all other quantities at their initial values) such that the sum of the torques produced by the blue and the red forces is zero, i.e. that there is no net force, and hence no net angular acceleration, and hence no rotational motion of the bar?  • Fred = 10N • Torque = force times lever arm. If the lever arm is half as long, we need twice as much force: f= 10 N

8. Newton II for Rotation

9. Work and Rotational Kinetic Energy • W = ∫ τ dθ • P = dW/dt = τω

10. Lecture 24: General Rotations

11. Worksheet: Torque and angular acceleration

12. Rotation plus Translation • Rolling is a combination of motion of the COM and rotation about the COM • Point of contact remains stationary, while point on top of wheel moves with twice the velocity of the COM.

13. Rolling as pure Rotation • Rolling can also be viewed as a pure rotation around the point of contact with floor • Need parallel axis theorem to calculate correct MOI • Two parts represent contributions from rotation and translation to KE

14. Forces of rolling • Friction is static, since point of contact is stationary. • Rolling down a ramp: • Can calculate is in linear coordinates: static friction and mg sin β determine linear acc. • Torque due to static friction acting at radius R determines angular acc.: torque = I α

15. Torque as a Vector

16. Pre-lecture Exercise (11.1 – 11.6) • What is the direction of the torque produced by a force pointing in the SW direction and at a point 2m directly below the origin? • a. SW b. Down • c. NE d. NW • e. 45 degrees upward from West • f. None of the above. • Answer: torque = r x F, where r is in –k, and F in –i – j direction, so –i+j or NW or d)

17. A ladybug sits at the outer edge of a merry-go-round, that is turning and slowing down due to a force exerted on its edge. At the instant shown, the torque on the disc is pointing in … -z direction -y direction +y direction +z direction z y F x

18. Reminder: Vector Product

19. What is AxA ? • Zero • A • -A • A2

20. Lecture 25: Angular Momentum

21. Angular Momentum

22. A ladybug sits at the outer edge of a merry-go-round, that is turning and slowing down due to a force exerted on its edge. The angular momentum of the bug is pointing in … -z direction -y direction +y direction +z direction z y F x

23. Pre-lecture Exercise (11.7-11.10) • What is the direction of the angular momentum (around the origin) of a particle located 2m directly above the origin (i.e. 2m along the z axis) with a momentum vector in the SW (i.e. –i–j) direction? • a. SW b. Down • c. NE d. NW • e. 45 degrees upward from West • f. None of the above. • Answer: l = r x p, where r is in k, and p in –i – j direction, so i-j or SE so f)

24. Angular Momentum of Rigid Body about fixed axis • L = sum over angular momenta of parts • Use l = r p = m r v = m r (r ω)  L = I ω

25. Dynamics Transliteration • Mass  Moment of inertia • Force  Torque • Momentum  Angular Momentum • Newton II • Momentum Conservation  Angular Momentum Conservation

26. Angular Momentum Conservation • Angular momentum is conserved if no external net torque is present • Demos: turntable + weights, turntable and bikewheel, bikewheel spinning on rope

27. Post-lecture Exercise (11.7 – 11.10) • In the sample problem on page 285, what are the magnitude and direction of the net angular momentum L about point O of the two-particle system if the velocity of particle two is reversed (180 degrees direction change)? The direction is going to be either out of the page (positive L) or into the page (negative L). • Answer: The only change is the sign of the vector l2, so L = (10 +8) kg m/s2= 18 kg m/s2

28. Lecture 26: Rotational Energy

29. Pre-lecture Exercise (11.10 – 11.12) • By what factor does the spinning volunteer’s period of rotation (p. 291) change if he is able to reduce is moment of inertia by a factor of 1.5? (Hint: your answer should be smaller than one if the period is reduced, and bigger than one if the period gets longer.) • Answer: I ω = const, so if I goes down by 1.5, ω goes up by 1.5, so the period goes down by 1.5, or f = 1/1.5 =2/3 = 0.6667.

30. Precession of a Gyroscope • Demo: Little Gyroscope • Demo: Bike Wheel on stick • Rate of precession becomes larger as wheel slows down

31. Gravitation

32. History • Ptolemy • Copernicus • Brahe • Galilei • Kepler • Newton

33. From Galileo to Newton - the Birth of Modern Science 1609 1687

34. Precursor: Nicolas Copernicus (1473–1543) • Rediscovers the heliocentric model of Aristarchus • Planets on circles • needs 48(!!) epicycles to explain different speeds of planets • Not more accurate than Ptolemy Major Work :De Revolutionibus Orbium Celestium (published posthumously)

35. Geocentric vs Heliocentric: How do we know? • Is the Earth or the Sun the center of the solar system? • How do we decide between these two theories? • Invoke the scientific methods: • both theories make (different) predictions • Compare to observations • Decide which theory explains data

36. Phases of Venus Heliocentric Geocentric

37. Sunspots • MPEG video from Galileo Project (June 2 – July 8, 1613)

38. Galileo and his Contemporaries • Elizabeth I. (1533-1603) – Queen of England • Tycho Brahe (1546-1601) – Danish Astronomer • Francis Bacon (1561-1626) – English Philosopher • Shakespeare (1564- 1616) – Poet & Playwright • Galileo Galilei (1564-1642) – Italian PAM • Johannes Kepler (1571-1630) – German PAM • Rene Descartes (1596 - 1650) – French PPM • Christiaan Huygens (1629-1695) – Dutch PAM • Isaac Newton (1643-1727) – English PM • Louis XIV (1638-1715) – French “Sun King”

39. ObservationsPhenomenology/TheoryExperiment  Data Predictions test predictions Tycho Brahe Johannes Kepler Galileo Galilei

40. Key question: How are things happening? Major Works: Harmonices Mundi (1619) Rudolphian Tables (1612) Astronomia Nova Dioptrice Johannes Kepler–The Phenomenologist Johannes Kepler (1571–1630)

41. Kepler’s First Law The orbits of the planets are ellipses, with the Sun at one focus

42. Ellipses a = “semimajor axis”; e = “eccentricity”

43. Lecture 27: Gravitation • Piazza Convocation  Short Lab! • Starry Monday tonight 7pm, here (Sci 238)

44. Kepler’s Second Law An imaginary line connecting the Sun to any planet sweeps out equal areas of the ellipse in equal times

45. Why is it warmer in the summer than in the winter in the USA? • Because the Earth is closer to the Sun • Because the Sun is higher in the sky in the summer • None of the above

46. Axis Tilt – earth as gyroscope • The Earth’s rotation axis is tilted 23½ degrees with respect to the plane of its orbit around the sun (the ecliptic) • It is fixed in space  sometimes we look “down” onto the ecliptic, sometimes “up” to it Rotation axis Path around sun

47. The Seasons • Change of seasons is a result of the tilt of the Earth’s rotation axis with respect to the plane of the ecliptic • Sun, moon, planets run along the ecliptic

48. Animation • TeacherTube video

49. Position of Ecliptic on the Celestial Sphere • Earth axis is tilted w.r.t. ecliptic by 23 ½ degrees • Equivalent: ecliptic is tilted by 23 ½ degrees w.r.t. equator! •  Sun appears to be sometime above (e.g. summer solstice), sometimes below, and sometimes on the celestial equator

50. The vernal equinox happens when the sun enters the zodiacal sign of Aries, but is actually located in the constellation of Pisces.